Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10)(11,12)(13,14)(15,16)(17,18)(19,25)(20,30)(21,23)(22,27)(24,28)(26,31)(29,32);(1,2)(3,5)(4,6)(7,8)(10,14)(11,15)(17,19,24,21)(18,25,28,23)(20,22,31,29)(26,32,30,27);(1,2,4,6)(3,7,8,5)(9,11,13,15)(10,12,14,16)(17,20,24,31)(18,26,28,30)(19,29,21,22)(23,32,25,27);(1,3,4,8)(2,5,6,7)(9,12)(10,15)(11,14)(13,16)(17,21,24,19)(18,23,28,25)(20,29,31,22)(26,27,30,32);(1,3)(2,5)(4,8)(6,7)(17,22,24,29)(18,27,28,32)(19,20,21,31)(23,26,25,30);(1,2)(3,5)(4,6)(7,8)(17,23)(18,19)(20,27)(21,28)(22,26)(24,25)(29,30)(31,32)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8898092);